More blades hush a propeller only while its tips are slow

A wire sings with the force on it, not the flow round it followed Curle’s extension of Lighthill’s analogy — the eighth-power law of the sound that only leaves — to a flow with a body in it. The body’s surface turns the flow’s own quadrupoles into a dipole, and the dipole’s strength is the fluctuating force the fluid exerts on the body: a wire’s tone is its oscillating lift, its sound power growing as the sixth power of the wind’s speed, louder than the wake’s own eighth-power noise. That surface was held still, and its force had to fluctuate to radiate. A steady force on a still body makes no sound at all.

It ended on the case where the force does not fluctuate and the surface moves. A propeller’s blades carry a thrust and a torque that are steady in the blade’s own frame, and they carry them round a circle at a sizable fraction of the speed of sound. Ffowcs Williams and Hawkings extended Curle’s formula to surfaces in arbitrary motion in 1969, and for a compact force moving on a circle the result had been found already, by Gutin in 1936. The essay’s question was how fast such a rotor’s tone grows with its tip’s Mach number, as the Doppler factor of the sound now is the source then approaches its singularity.

Gutin’s answer is a Bessel function, and its order is what makes the result useful. Each harmonic of the blade-passing tone is carried by a Bessel function whose order equals the harmonic’s number of cycles per revolution, and such a function is exponentially small until the tip’s speed towards the listener nears the speed of sound. More blades raise that order. So they silence a slow propeller dramatically and a fast one hardly at all.

A force that only moves #

A blade carries two steady forces. Its thrust pushes the air aft along the axis; its drag, the reaction to the torque the engine supplies, drags the air round with the rotation. With B blades the forces are concentrated here at an effective radius of 0.8 of the tip — the radius at which a blade’s loading is usually taken to act — and each blade carries a B-th of each. Seen from a fixed point, a steady force at a moving point is a force whose position changes, and a dipole at a changing position radiates: the pressure it makes at a listener depends on where it was when the sound left it, which is a different place each moment.

Because the blades are equally spaced and identical, the pattern repeats B times a revolution, and the sound comes only at harmonics of the blade-passing frequency $B\Omega$. Gutin’s far-field result for the m-th harmonic, at distance $r$ and angle $\theta$ from the forward axis, is

$$p_m = \frac{mB\Omega}{2\sqrt{2}\,\pi c\, r}\,\left|\,T\cos\theta - \frac{Qc}{\Omega R_e^2}\,\right|\, J_{mB}\!\left(mB\, M \sin\theta\right),$$

with $T$ and $Q$ the total thrust and torque, $R_e$ the effective radius, $M = \Omega R_e/c$ its Mach number, and $p_m$ the harmonic’s root-mean-square pressure. The bracket is the steady loading resolved towards the listener — the thrust along the axis, the torque’s force round the circle weighted by how fast the circle turns. The Bessel function is the rotation.

It was checked here by a second route that shares nothing with the Bessel function. Each blade’s force was followed round its circle in the time domain; for every emission time the far-field pressure it sends — the time derivative of the force resolved towards the listener over the distance and the Doppler factor — was computed at the time it arrives, one revolution was resampled on a uniform grid of arrival times, and its harmonics were read off. The two routes agree to eight parts in a hundred thousand, for the first four harmonics at three angles. A force held still — the same propeller at a millionth of a revolution a minute — radiates nothing, to rounding error.

The case is a light aircraft’s propeller in its climb: two blades, 1.88 metres across, 2,700 rpm, a tip Mach number of 0.78, giving 2 kilonewtons of thrust from 127 kilowatts, heard thirty metres away. When its speed is changed below, its thrust and its shaft power are held, so the torque falls as the rotation speeds up.

The tone rises steeply with the tip’s speed #

The first figure is the level of the first four harmonics, each at the angle where it is loudest, against the tip Mach number. At the climb’s 0.78 the fundamental is 98.2 decibels at thirty metres and the fourth harmonic 88.6. At a tip Mach number of 0.4 the fundamental is 89.8 — only 8 decibels lower, since it is a low-order Bessel function — but the fourth harmonic has fallen to 51. And as the tip nears the speed of sound the harmonics close up: at 0.95 the fundamental is 101 decibels and the fourth harmonic 98.9.

That is the shape of the result throughout. The higher the harmonic, the more steeply it rises with the tip’s speed, and near a tip Mach number of one they all become comparable. A slow propeller’s tone is nearly a pure note at the blade-passing frequency; a fast one’s is rich in harmonics, a harsh buzz, and its sound has moved up the spectrum towards the frequencies to which ears are most sensitive.

A Bessel function of its own order #

The second figure is the mechanism by itself. The Bessel function’s order is $n = mB$, the number of cycles per revolution of the harmonic, and its argument is $n$ times $x = M\sin\theta$ — the greatest speed towards the listener of a point on the effective circle, as a fraction of the speed of sound. For every order the function climbs towards a value of a few tenths as $x$ approaches one; for small $x$ it is tiny, and the higher the order the tinier. At $x = 0.4$ the second-order function is 0.076, the eighth-order 0.0008, the sixteenth-order $3\cdot10^{-6}$. At $x = 0.9$ they are 0.31, 0.15 and 0.084.

The reason is cancellation round the circle, the same thing the sound is what does not cancel found in a quadrupole. The n-th harmonic of a source spread round a circle has n wavelengths of phase around its circumference, while the sound it sends towards the listener has a phase that varies round the circle only by the source’s distance towards the listener, $kR\sin\theta$ — which is $n$ times $x$. When $x$ is less than one, the circle’s own phase pattern is finer than anything the outgoing wave can carry, and the contributions from round the circle cancel almost exactly: the field near the circle is evanescent, decaying away from it rather than propagating. Only as $x$ nears one — as some part of the circle moves towards the listener at nearly the speed of sound — do the phases line up and the cancellation fail. Debye’s asymptotic form makes it quantitative: below $x = 1$ the function falls as a fixed factor per unit of order, about 0.5 per order at $x = 0.4$ and 0.97 per order at $x = 0.9$.

That is the Doppler factor’s singularity seen in frequency. In time, a point on the circle approaching the listener at nearly the speed of sound has its emissions compressed into a narrow interval of arrival times, and a narrow pulse has high harmonics; in frequency, the same compression is the phase-matching that lets a high-order Bessel function become large.

Where it is loudest, and where it is silent #

The third figure draws the tone round the propeller. The Bessel function’s argument is largest in the plane of the disc, at 90°, where the circle’s motion points straight at the listener twice a revolution, and falls to zero on the axis, where the circle is always the same distance away and nothing changes: a propeller’s steady loading is silent straight ahead and straight behind.

The bracket moves the peak and puts a null in front. Ahead of the disc the thrust’s contribution, which goes as the cosine of the angle, opposes the torque’s; at the climb they cancel at 61° from the forward axis, where both harmonics vanish. Behind the disc they add. So the loudest direction is a little behind the plane of rotation: 118° for the fundamental and 109° for the second harmonic at a tip Mach number of 0.78. The dial shows the pattern moving with the tip’s speed. At 0.4 there is no null — the torque term, divided by a slow rotation, outweighs the thrust’s at every angle — and the fundamental peaks at 104°; at 0.9 the null has moved to 69° and the peak to 120°, and the second harmonic is nearly as loud as the fundamental.

More blades, the same load #

The fourth figure answers the practical question. Keep the thrust, the power and the diameter, and divide the load among two, three, four or six blades. Each blade carries less, but the total force round the circle is the same — the same thrust the most a disc can take prices in momentum — so the bracket in Gutin’s formula is unchanged; what changes is the fundamental’s order, $n = B$, and with it the Bessel function, and the frequency, which rises with $B$.

At a tip Mach number of 0.4 six blades are 25 decibels quieter than two. At the climb’s 0.78 they are 5.8 decibels quieter. At 0.95, 1.1 decibels. The gain from more blades is almost entirely the Bessel function’s order, and the Bessel function is exponentially small in its order only while its argument is well below the order. More blades hush a propeller only while its tips are slow. Near the speed of sound every order is large, the blades’ number hardly matters, and the only lever left is the tip speed itself.

That is why quiet propellers combine many blades with slow tips — the six- and eight-bladed propellers of large turboprops turning slowly — and why a propeller whose tips approach the speed of sound cannot be made quiet by adding blades. The same arithmetic sets the price of a geared engine: slowing the propeller to bring its tips down buys back, at the same thrust and power, the order-by-order suppression that the extra blades then multiply. Thrust and power alone do not set the tone; a big slow push is quieter as well as more efficient than a small fast one, for a reason the momentum theory of a disc does not contain.

What the ear makes of it #

The fourth figure compared fundamentals, and two corrections stand between it and what a listener hears. The first is that the fundamental is not the whole tone. A two-bladed propeller near the speed of sound has many comparable harmonics, at two, four, six and more cycles a revolution, while a six-bladed one has only every third of them; summed in energy over the first twelve harmonics, six blades are 25 decibels quieter than two at a tip Mach number of 0.4, 8.3 at the climb’s 0.78 and 5.6 at 0.95 — more than the fundamentals alone suggested near the speed of sound, because there the two-bladed propeller’s extra harmonics count.

The second is the ear. More blades raise every frequency: at the climb the two-bladed fundamental is at 90 hertz and the six-bladed one at 269, and the A-weighting, which follows the ear’s falling sensitivity at low frequencies, discounts the first by 20.7 decibels and the second by 8.0. The fifth figure sums the harmonics with each one weighted at its own frequency. Weighted, six blades are 11 decibels quieter than two at a tip Mach number of 0.4, not 25, and 4.6 quieter at both 0.78 and 0.95. The Bessel function’s suppression is real and large, but part of what it removes was sound the ear barely heard, and part of what the extra blades add is moved to where the ear hears it well. A design comparison quoted in unweighted levels flatters the many-bladed propeller at low tip speeds by a factor of two in decibels.

A blade passage, in time #

The sixth figure is the time-domain integration itself, the check’s own output, over one blade passage. At a tip Mach number of 0.5 the pressure is nearly a sine at the blade-passing frequency, with a peak of 1.3 pascals at thirty metres. At 0.9 it is a sharp pulse, with a peak of 8.8 pascals, arriving once a blade passage when the blade’s motion points most nearly at the listener and its emissions crowd together in arrival time. The pulse is the time-domain form of the harmonics that rise together in the first figure: a narrow, tall pulse is a spectrum with many harmonics of nearly equal size.

The pulse’s shape is also where the linear theory starts to strain. A pressure pulse of nearly ten pascals arriving over a fraction of a millisecond is still an acoustic wave at thirty metres, but closer to the disc, and as the tip Mach number passes one, the pulse steepens into the shock that every compression becomes — the crack of a propeller or a helicopter rotor whose tips are supersonic.

The convention: rms pressure at thirty metres #

Levels are root-mean-square sound pressure levels of each harmonic, in decibels re 20 micropascals, in the far field, scaled to thirty metres by the inverse distance. “Loudest” is the greatest over the angle from the forward axis. The tip Mach number is the blade tip’s rotational speed over the speed of sound, 340 metres a second; the forward flight speed is zero in the formula, so the climb’s own speed is left out. The loads act at 0.8 of the tip radius. When the tip speed or the blade count is changed, the thrust and the shaft power are held.

What the calculation leaves out #

Thickness noise. The blade also displaces air as it moves — a monopole from its volume, which for thin blades is smaller than the loading noise at moderate tip speeds and comparable to it near the speed of sound. Ffowcs Williams and Hawkings’ equation has a term for it, not computed here.

Unsteady loading. A real blade’s thrust fluctuates as it passes through the wakes of the wing, the engine’s inflow distortion and the gusts of turbulence, and those fluctuations radiate at every harmonic without the Bessel function’s suppression. At low tip speeds this broadband and unsteady loading noise is often louder than Gutin’s steady tone, which is exactly why the steady tone’s suppression matters less in practice than the figures might suggest. A blade that flies through what it shed describes one source of it.

Forward flight. A propeller moving forward through the air changes the relative motion of its blades and the listener, and the Doppler factor gains a forward component; Garrick and Watkins extended Gutin’s result to it in the 1950s.

The loading’s distribution. Every blade’s forces are put at one radius. Real loads are spread along the blade, and the outer parts, moving faster, radiate more; a radial integral of Gutin’s formula with the blade’s own loading would weight the tip more heavily than the 0.8 used here.

Nonlinear propagation. At supersonic tips the radiated pulse steepens into a shock, which the linear acoustic analogy does not describe.

Who found it #

Gutin derived the steady-loading tone of a propeller in 1936, from the field of rotating point forces; Deming and Garrick extended the theory to thickness noise and forward flight in the late 1930s and 1950s; Lowson’s formula for the field of a point force in arbitrary motion, in 1965, and Ffowcs Williams and Hawkings’ equation, in 1969, put it in the general frame the earlier essays use. The exponential dependence on blade number at low tip speeds is a standard result in propeller acoustics; the figures here compute it for one stated propeller and check the formula against an independent time-domain integration.

How the tone was checked #

The Bessel function is computed by its power series below its order, where the integral representation loses the tiny values to rounding, and by the integral above it; where both are good they agree to a few parts in $10^{15}$, and the second-order function at one matches its tabulated value to the seventeenth decimal. Gutin’s formula and the time-domain integration agree to eight parts in a hundred thousand for four harmonics at three angles, and a force that does not turn radiates $10^{-19}$ pascals.

Still open: the tone a turbulent inflow puts back #

The Bessel function’s suppression applies to the steady loading only. A propeller ingesting turbulence — at a static test, behind a wing, close to the ground — sees its thrust fluctuate on each blade as it cuts through eddies, and an unsteady load at a harmonic $s$ of the rotation radiates through a Bessel function of order $n - s$ rather than $n$, which can be small. The next calculation would give each blade a loading fluctuation of stated spectrum, from a turbulent inflow of stated scale, and ask at what tip Mach number the unsteady loading’s tone overtakes the steady one for two and for six blades — which decides whether the many-bladed propeller’s advantage survives the turbulence it actually flies in.